The purpose of this material is to show the relationships among the
solenoids terms which arise in meteorological analysis. A solenoid term is of the
form ∇f×∇g,
where f and g are scalar field variables. The remarkable fact is that when f and g are
field variables such as temperature, pressure, density, specific volume and entropy the
solenoid terms are proportional.

The magnitude of such a term is equal
to |∇f||∇g|sin(φ),
where φ is the angle between the two gradients. The solenoid vector lies in the line
intersection of the two surfaces of constant levels of f and g.

The blue vector perpendicular to the blue-outlined plane is the gradient vector for the
f field. The red vector perpendicular to the red-outlined plane is the gradient vector for the
g field. The green vector in the line of intersection of the two planes is their vector cross
product. If the two constant
surfaces coincide then the angle between the gradients is zero and the solenoid term
vanishes. This condition prevails when the atmosphere is barotropic; i.e., when the
atmospheric density is a function only of the pressure.

Derivation of the Equation for the Time Rate of Change of Vorticity

For an inertial frame of reference
the equations of motion for a
parcel of air are, in vector form,:

dv/dt = -(1/ρ)∇p -
gk + f

where v is the velocity vector, ρ the density, p pressure, g the
acceleration due to gravity, k the unit
vertical vector and f the
vector of friction forces.

The pressure gradient term

-(1/ρ)∇p

is especially important.

This term can be put into an interesting form by noting that from the
definition of potential temperature θ:

ln(θ) = ln(T) + κln(p0) - κln(p)

and when the gradient operator ∇
is applied to this equation the result is

dv/dt = T∇s − ∇h
−gk + f

The motion-following derivative dv/dt is composed of an instaneous
rate of change at a point and an advection term; i.e.,

dv/dt = ∂v/∂t + v·∇v

The advection term v·∇v can be
expressed1 as

∇(v2/2) − v×(∇×v)

but ∇×v is
just the vorticity vector qsov·∇v = ∇(v2/2) - v×q

Thus the equations of motion for the atmosphere can be
expressed in vector form as

(1) ∂v/∂t =
v×q +
T∇s − ∇(v2/2 + h + gz) + f

The curl operator ∇× can be
applied to
this equation. The curl of any gradient of a scalar field vanishes;
i.e.,
∇×∇γ=0 for any
scalar field γ because of the equality of cross derivatives.
Therefore under the curl operation ∇(v2/2 + h + gz) vanishes.

Also, because the curl of a curl vanishes,

∇×(T∇×s) = ∇T×∇s.

The result of applying the curl operator to the left-hand side of the above
equation of motion (1) and taking into account the interchangeability of the time and
space derivatives is

∇×(∂v/∂t)
= ∂(∇×v)/∂t
= ∂q/∂t

Equating this to the result of applying the curl operation to the right-hand side of the
equation (1) gives

∂q/∂t
= −∇×(v×q)
+ ∇T×∇s
+ ∇×f

This form of the vorticity equation points out the role of the
intersection or non-intersection of the isothermal surface and the
isoentropic surface through the term
∇T×∇s.

Note that since ∇s = cp∇T/T - R∇p/p

∇T×∇s =
∇T×(cp∇T/T - R∇p/p)
= −∇T×(R∇p/p) =
−(R/p)(∇T×∇p)

Thus the ∇T×∇s term in the vorticity
equation can be replaced by a term involving
∇p×∇ρ. Generally all
of these cross product terms, called solenoid terms, are proportional and they all vanish
when the atmosphere is barotropic; i.e.; when ∇ρ
always has the same direction as ∇p.

Combining the last two derivations gives

∇T×∇s = (RT/pρ)(∇p×∇ρ)
= (1/ρ²)∇p×∇ρ

The −∇×(v×q) term in the vorticity equation
reduces2 to

q(∇·v)
+ [(v·∇)q
− (q·∇)v]

These two terms are known as the divergence term and the twisting term, respectively.

Not much can be done analytically with the friction term except note that it is directed
opposite to the velocity vector. It is of a low order of magnitude than the other terms and is
usually neglected in the analysis.

The term vorticity in meterology usually refers to the vertical component of the
vorticity vector. The vertical component of a vector may be obtained analytically by take
the dot product of the vector with the unit vector in the vertical direction
k; i.e., ζ = k·q. Thus the
time rate of change of vorticity ζ is given by

dζ/dt = ζ(∇·v)
+ k·[(v·∇)q
−(q·∇)v]
+ (1/ρ²)k·(∇p×∇ρ)

The terms on the right-hand side of the equation are known, respectively, as the divergence term, the twisting or
tilting term and the solenoid term.